Endomorphisms - old and not so old Endomorphisms - old and not so old Joachim Cuntz Copenhagen 2019
A unique, even bizarre, institution in West Philadelphia since 1950, the Divine Tracy is up for sale. The asking price for the 140-room hotel is $10 million. In a city where hotels boast what makes them luxurious – from the thread count in the bed sheets to spa services for pets – the Divine Tracy stands out for the niche it serves. For a flat $50 a night rate, guests can enjoy austere but consistent accommodations in which men and women are housed on separate floors. Guests must adhere to a so-called International Modest Code, which was developed by Father Divine, the spiritual leader of a Christian-based ministry called the Palace Mission. The code, which sets high standards for behavior, provides that guest do not smoke, drink alcohol, use obscenities, vulgarity, profanity, receive gifts, presents, tips or bribes. There is no eating food in the rooms and the dress code bespeaks of modesty. Women are not permitted to wear pants, shorts or miniskirts; men must not don sleeveless shirts, have their shirts untucked or wear shoes sans socks. In addition, there must be no “undue mixing of the sexes” but men and women may converse in the lobby. Rooms do not have televisions. (Note: At the time the rate was $50 for an entire week.)
SIMPLE c* -ALGEBRAs GENERATED BY ISOMETRIES Joachim Cuntz N r. 27 · / F ebru a r 1977
The C*-algebra A [ α ] Let K be a compact abelian group. Typical examples: K = T n K = ( Z / n ) N K = lim T ← − z �→ z n We consider an endomorphism α of K satisfying ◮ α is surjective ◮ Ker α is finite ◮ � n Ker α n is dense in K .
The C*-algebra A [ α ] Let K be a compact abelian group. Typical examples: K = T n K = ( Z / n ) N K = lim T ← − z �→ z n We consider an endomorphism α of K satisfying ◮ α is surjective ◮ Ker α is finite ◮ � n Ker α n is dense in K . Then α preserves Haar measure on K and therefore induces an isometry s α on L 2 K . Also C ( K ) act as multiplication operators on L 2 K .
The C*-algebra A [ α ] Let K be a compact abelian group. Typical examples: K = T n K = ( Z / n ) N K = lim T ← − z �→ z n We consider an endomorphism α of K satisfying ◮ α is surjective ◮ Ker α is finite ◮ � n Ker α n is dense in K . Then α preserves Haar measure on K and therefore induces an isometry s α on L 2 K . Also C ( K ) act as multiplication operators on L 2 K . Definition We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α .
The C*-algebra A [ α ] Let K be a compact abelian group. Typical examples: K = T n K = ( Z / n ) N K = lim T ← − z �→ z n We consider an endomorphism α of K satisfying ◮ α is surjective ◮ Ker α is finite ◮ � n Ker α n is dense in K . Then α preserves Haar measure on K and therefore induces an isometry s α on L 2 K . Also C ( K ) act as multiplication operators on L 2 K . Definition We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α .
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian.
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian. By Fourier transform A [ α ] then is isomorphic to the C*-subalgebra A [ ϕ ] of L ( ℓ 2 G ) generated by the left regular representation λ of G and by the isometry s ϕ defined on ℓ 2 G by s ϕ ( δ g ) = δ ϕ ( g ) where ϕ = ˆ α is the dual endomorphism of G = � K .
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian. By Fourier transform A [ α ] then is isomorphic to the C*-subalgebra A [ ϕ ] of L ( ℓ 2 G ) generated by the left regular representation λ of G and by the isometry s ϕ defined on ℓ 2 G by s ϕ ( δ g ) = δ ϕ ( g ) where ϕ = ˆ α is the dual endomorphism of G = � K . This dual construction of A [ ϕ ] can be generalized to an endomorphism ϕ of a not necessarily abelian discrete group G satisfying ◮ ϕ is injective ◮ G /ϕ G is finite ◮ � ϕ n ( G ) = { e }
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian. By Fourier transform A [ α ] then is isomorphic to the C*-subalgebra A [ ϕ ] of L ( ℓ 2 G ) generated by the left regular representation λ of G and by the isometry s ϕ defined on ℓ 2 G by s ϕ ( δ g ) = δ ϕ ( g ) where ϕ = ˆ α is the dual endomorphism of G = � K . This dual construction of A [ ϕ ] can be generalized to an endomorphism ϕ of a not necessarily abelian discrete group G satisfying ◮ ϕ is injective ◮ G /ϕ G is finite ◮ � ϕ n ( G ) = { e } This situation has been studied by Ilan Hirshberg who showed that A [ ϕ ] is simple if G is amenable and ϕ n ( G ) is normal in G for all n .
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian. By Fourier transform A [ α ] then is isomorphic to the C*-subalgebra A [ ϕ ] of L ( ℓ 2 G ) generated by the left regular representation λ of G and by the isometry s ϕ defined on ℓ 2 G by s ϕ ( δ g ) = δ ϕ ( g ) where ϕ = ˆ α is the dual endomorphism of G = � K . This dual construction of A [ ϕ ] can be generalized to an endomorphism ϕ of a not necessarily abelian discrete group G satisfying ◮ ϕ is injective ◮ G /ϕ G is finite ◮ � ϕ n ( G ) = { e } This situation has been studied by Ilan Hirshberg who showed that A [ ϕ ] is simple if G is amenable and ϕ n ( G ) is normal in G for all n . We will however consider only the case where G is abelian (and ϕ satisfies the three conditions above).
Definition. We denote by A [ α ] the sub-C*-algebra of L ( L 2 K ) generated by C ( K ) together with s α . Description of A [ α ] in the Fourier transform picture It is important to describe A [ α ] = C ∗ ( C ( K ) , s α ) also in the dual picture: Let G = � K denote the dual group which is discrete abelian. By Fourier transform A [ α ] then is isomorphic to the C*-subalgebra A [ ϕ ] of L ( ℓ 2 G ) generated by the left regular representation λ of G and by the isometry s ϕ defined on ℓ 2 G by s ϕ ( δ g ) = δ ϕ ( g ) where ϕ = ˆ α is the dual endomorphism of G = � K . This dual construction of A [ ϕ ] can be generalized to an endomorphism ϕ of a not necessarily abelian discrete group G satisfying ◮ ϕ is injective ◮ G /ϕ G is finite ◮ � ϕ n ( G ) = { e } This situation has been studied by Ilan Hirshberg who showed that A [ ϕ ] is simple if G is amenable and ϕ n ( G ) is normal in G for all n . We will however consider only the case where G is abelian (and ϕ satisfies the three conditions above).
Under these conditions the algebra A [ α ] = A [ ϕ ] always contains a canonical commutative subalgebra D which is a Cartan subalgebra. It is generated by the translates λ g s n ϕ s ⋆ n g of the range projections s n ϕ s ⋆ n ϕ λ ⋆ ϕ . The spectrum of D is the ϕ -adic completion ¯ − G /ϕ n G of G . This G = lim ← completion is actually a compact abelian group. Since G is dense in ¯ G , the action of G on ¯ G is minimal and the crossed product D ⋊ G is simple. We denote this subalgebra by B . It is also immediate that B has a unique tracial state.
Under these conditions the algebra A [ α ] = A [ ϕ ] always contains a canonical commutative subalgebra D which is a Cartan subalgebra. It is generated by the translates λ g s n ϕ s ⋆ n g of the range projections s n ϕ s ⋆ n ϕ λ ⋆ ϕ . The spectrum of D is the ϕ -adic completion ¯ − G /ϕ n G of G . This G = lim ← completion is actually a compact abelian group. Since G is dense in ¯ G , the action of G on ¯ G is minimal and the crossed product D ⋊ G is simple. We denote this subalgebra by B . It is also immediate that B has a unique tracial state. Theorem (Cuntz-Vershik) A [ α ] is simple and purely infinite. It can be described as a universal C*-algebra with a natural set of generators and relations (as a consequence A [ α ] is also isomorphic to a crossed product B ⋊ N ).
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